Question: $g(r)=r^2-6r-55$ 1) What are the zeros of the function? Write the smaller $r$ first, and the larger $r$ second. $\text{smaller }r=$
Answer: To find the zeros of the function, we need to solve the equation $g(r)=0$. We can do that by factoring $g(r)$. $\begin{aligned} r^2-6r-55&=0 \\\\ (r-11)(r+5)&=0 \\\\ r-11=0&\text{ or }r+5=0 \\\\ r={11}&\text{ or }r={-5} \end{aligned}$ There are many ways to find the vertex. We will do it by using the fact that the $r$ -coordinate of the vertex is exactly between the two zeros. $\begin{aligned} \text{vertex's }r\text{-coordinate}&=\dfrac{({11})+({-5})}{2} \\\\ &={3} \end{aligned}$ Now we can find the vertex's $y$ -coordinate by evaluating $g({3})$ : $\begin{aligned} g({3})&=({3})^2-6({3})-55 \\\\ &=9-18-55 \\\\ &=-64 \end{aligned}$ In conclusion, $\begin{aligned} \text{smaller }r&=-5 \\\\ \text{larger }r&=11 \end{aligned}$ The vertex of the parabola is at $(3,-64)$